Finding the Median (Obliviously) with Bounded Space

TL;DR

This paper establishes a nearly tight lower bound on the time complexity of oblivious algorithms for median finding with bounded space, highlighting fundamental limits in space-time trade-offs for this problem.

Contribution

It proves a new lower bound on the time complexity of oblivious median finding algorithms using limited space, extending understanding of computational constraints.

Findings

Any oblivious median-finding algorithm with space S requires at least Omega(n log log_S n) time.

The bound applies to determining the median's parity, not just its value.

Implications include size lower bounds for read-once branching programs and complexity class separations.

Abstract

We prove that any oblivious algorithm using space $S$ to find the median of a list of $n$ integers from $\{1,...,2n\}$ requires time $\Omega(n \log\log_S n)$. This bound also applies to the problem of determining whether the median is odd or even. It is nearly optimal since Chan, following Munro and Raman, has shown that there is a (randomized) selection algorithm using only $s$ registers, each of which can store an input value or $O(\log n)$-bit counter, that makes only $O(\log\log_s n)$ passes over the input. The bound also implies a size lower bound for read-once branching programs computing the low order bit of the median and implies the analog of $P \ne NP \cap coNP$ for length $o(n \log\log n)$ oblivious branching programs.